10 J^ THE DETONATION PROCESS 



these quantities, which include the effect of motion of the boundary, 

 must therefore also be equal, giving 



dPn dP dUn du 

 —r- = —r' —r = -v> at r = a 

 dt dt dt dt 



The fundamental variable of the Kirkwood-Bethe theory is the function 

 6r(r, t), constant values of which are propagated with a velocity c such 

 that 



dG , .aG „ 

 dt dr 



and it is therefore necessary to relate G to the boundary conditions. 

 From its definition, G satisfies the differential equation 



dG = d Ir L + ^\ =L + "0 dr + rdL + |'^ 



where the enthalpy co satisfies the fundamental thermodynamic relation 

 doj = TdS + dP/p. The numerical calculations of section 2.7 show 

 that the entropy term is small, and setting dS = gives 



=(-r) 



dG = { CO + ^]dr -\-^dP + rudu 



where oi = & f dp/ p. Substitution in the propagation equation for G 

 gives 



This equation is valid in both the water and gas products if the partial 

 derivatives are evaluated in the medium considered. 



The basic dynamical equations of motion and continuity must be 

 true in either medium, and in the water they are 



The equality of total derivatives at the boundary is of course not true 

 of the partial derivatives if the boundary moves, and the next step is to 

 solve for the desired total derivatives by elimination of the partial 

 derivatives. For the water this gives 



